Multifractality in random networks with power-law decaying bond strengths

In this paper we demonstrate numerically that random networks whose adjacency matrices ${\bf A}$ are represented by a diluted version of the Power--Law Banded Random Matrix (PBRM) model have multifractal eigenfunctions. The PBRM model describes one--dimensional samples with random long--range bonds. The bond strengths of the model, which decay as a power--law, are tuned by the parameter $\mu$ as $A_{mn}\propto |m-n|^{-\mu}$; while the sparsity is driven by the average network connectivity $\alpha$: for $\alpha=0$ the vertices in the network are isolated and for $\alpha=1$ the network is fully connected and the PBRM model is recovered. Though it is known that the PBRM model has multifractal eigenfunctions at the critical value $\mu=\mu_c=1$, we clearly show [from the scaling of the relative fluctuation of the participation number $I_2$ as well as the scaling of the probability distribution functions $P(\ln I_2)$] the existence of the critical value $\mu_c\equiv \mu_c(\alpha)$ for $\alpha<1$. Moreover, we characterise the multifractality of the eigenfunctions of our random network model by the use of the corresponding multifractal dimensions $D_q$, that we compute from the finite network-size scaling of the typical eigenfunction participation numbers $\exp\left\langle\ln I_q \right\rangle$.